I have just a few more comments about the paper on long division by DavidKlein and Jim Milgram.

1. cost-benefit analysis.

K & M focus on the benefits of teaching the standard long divisionalgorithm. I believe they exaggerate the necessity of standard longdivision as the sole means to achieve several of the benefits they cite. Ialso believe that many of the benefits they outline are rarely achieved inpractice. They are theoretical, not real, goods. (Interestingly, thepractical justification for teaching long division has been renderedlargely obsolete by calculators. Anyone who has a practical need to do lotsof long division problems is very unlikely to use any paper-and-pencilalgorithm. K & M had to look elsewhere for reasons to retain the algorithm.)

But even if K & M are entirely correct in their claims about the benefitsof teaching standard long division, their analysis is incomplete becausethey have failed to examine the costs of teaching this algorithm. (There issome arm waving about methods for teaching long division to children, but,as far as I know, these ideas are speculative and have not been tested inreal classrooms. People have been trying to figure out a good way to teachstandard long division for a long time, and my guess is that if there wereone it would have been found out by now.)

It is impossible to judge whether it is worthwhile to teach the standardlong division algorithm unless the costs are examined. More generally,other approaches to long division, with their costs and benefits, shouldalso be examined. And this examination should not be purely a priori. Data,including student achievement data, should be considered.

I claim K & M seriously understimate the costs of teaching the standardlong division algorithm. I am not talking about the costs when thealgorithm is taught by an expert teacher under ideal conditions. I amtalking about the costs when ordinary teachers teach the algorithm undertypical conditions. Bringing all children to something like mastery takesmonths of time over several years. The approach typically taken to thestandard long division algorithm is highly procedural and syntactic -- insharp contrast to the conceptual approach K & M clearly favor -- whichengenders misunderstanding of what mathematics is and consequent distastefor the subject.

Perhaps K & M meant their piece as a correction to reformers like Leinwandwho have spent a lot of time identifying the high costs of teachingstandard long division and other paper-and-pencil algorithms. If so, Ibelieve they have over-corrected. A balanced assessment of costs andbenefits would have been more helpful.

2. mathematics and mathematics education.

It would be helpful if K & M could at least pretend not to think that allmath educators are idiots. It may be difficult for them to do this, butperhaps an atmosphere of mutual respect could be established thatfacilitate progress in our field.

Roughly 100 years ago, John Dewey pointed out that it's not enough toconsider the subject matter that is being taught. One must also take thestudent into account. No doubt K & M have vastly greater knowledge ofmathematics than most people in mathematics education. Certainly they knowmore mathematics than I do. But there are people in mathematics educationwho not only know some mathematics, but have actually taught school. Anddone research and read others' research. And written elementary school mathtextbooks. And know the history of mathematics education. The views of suchpeople are not irrelevant to the questions whether and how long divisionshould be taught. If K & M are uninterested in the discipline ofmathematics education, I don't blame them. All I ask is that they recognizethe limits of their expertise.

3. single-issue politics.

Probably what disturbs me most is the focus on long division as the sinequa non for a curriculum to be acceptable. Not only must paper-and-pencillong division be taught, but one certain algorithm must be taught. Anyalternative is anathema.